Engine Modeling - Bernoulli Model

Hydraulics

Pratima Bajpai, in Biermann's Handbook of Pulp and Paper (Third Edition), 2018

Bernoulli's Principle

Bernoulli's principle is valid for any fluid (liquid or gas); it is especially important to fluids moving at a high velocity. Its principle is the basis of venturi scrubbers, thermo compressors, aspirators, and other devices where fluids are moving at high velocities. It also explains cavitation in fluids (such as in valves and pumps). The sum of pressure (potential energy) and kinetic energy in any system is constant (i.e., energy is conserved if frictional losses are ignored). Thus when a fluid flows through areas of different diameters, there is a change in velocity. The change in velocity leads to a change in kinetic energy and so the pressure changes as well. A decreased pipe diameter means an increase in velocity and kinetic energy and a decrease in pressure. 

Drag and Lift

Yasuki Nakayama, in Introduction to Fluid Mechanics (Second Edition), 2018

9.5 Cavitation

According to Bernoulli's principle, as the velocity increases, the pressure decreases correspondingly. In the forward part on the upper surface of a wing section placed in a uniform flow as shown in Fig. 9.24, for example, the flow velocity increases while the pressure decreases.

 An aerofoil section inside the flow.

If a section of a body placed in liquid increases its velocity so much that the pressure there is less than the saturation pressure of the liquid, the liquid instantaneously boils, producing bubbles with cavities. This phenomenon is called cavitation. In addition, since gas dissolves in liquid in proportion to the pressure (Henry's law), as the liquid pressure decreases, the dissolved gas separates from the liquid into bubbles even before the saturation pressure is reached. When these bubbles are conveyed downstream where the pressure is higher, they are suddenly squeezed and abnormally high pressure develops.9 At this point, noise and vibration occur eroding the neighbouring surface and leaving on it holes small in diameter but relatively deep, as if made by a slender drill in most cases. These phenomena as a whole are also referred to as cavitation in a wider sense.

The blades of a pump or water wheel, or the propeller of a boat, are sometimes destroyed by such phenomena. They can develop on liquid-carrying pipe lines or on hydraulic devices and cause failures.

The saturation pressures at various temperatures are shown in Table 9.3, while the volume ratios of air soluble in water at 1 atm are shown in Table 9.4.

Table 9.3. Saturation Pressure for Water

Temp. (°C)	Pa	Temp. (°C)	Pa
0	608	50	12,330
10	1226	60	19,920
20	2334	70	31,160
30	4236	80	47,360
40	7375	100	101,320

Table 9.4. Solubility of Air in Water

Temp. (°C)	0	20	40	60	80	100
Air	0.0288	0.0187	0.0142	0.0122	0.0113	0.0111

When an aerofoil section is placed in a flow of liquid, the pressure distribution on its surface is as shown in Fig. 9.25. As the cavity grows, the upper pressure characteristic curve lowers while vibration, etc. grow. When the liquid pressure is low and the flow velocity is large, the cavity grows further. When it grows beyond twice the chord length, the flow stabilizes, with noise and vibration reducing. This situation is called supercavitation and is applied to the wings of a hydrofoil boat.

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Development of cavitation on an aerofoil section.

Let the upstream pressure not affected by the wing be p∞, the flow velocity U and the saturation pressure pv. When the pressure at a point on the wing surface or nearby has reached pv, cavitation develops. The ratio of p∞ − pv to the dynamic pressure is expressed by the following equation:

(9.35)$k_d=\frac{p_{\mathit{\infty }}-p_v}{\rho U^2/2}$

kd in this equation is called the cavitation number. When kd is small, cavitation is likely to develop.

Engine Modeling

Daiheng Ni, in Traffic Flow Theory, 2016

19.3.3 Model III: Bernoulli Model

This model is based on the Bernoulli principle, which states that for an ideal fluid (e.g., air) on which no external work is performed, an increase in velocity occurs simultaneously with a decrease in pressure or a change in the fluid’s gravitational potential energy. When the fluid flows through a pipe (e.g., the intake manifold) with a constriction (e.g., the throttle) in it, the fluid velocity at the constriction must increase in order to satisfy the equation of continuity, while its pressure must decrease because of conservation of energy. The limiting condition of this effect is choked flow, where the mass flow rate is independent of the downstream pressure (e.g., in the combustion chamber), depending only on the temperature and pressure on the upstream side of the constriction (e.g., the atmosphere). The physical point at which the choking occurs is when the fluid velocity at the constriction is at sonic conditions or at a Mach number (the ratio of fluid velocity and sound speed) of 1. With the above knowledge, the Bernoulli engine model is developed as follows.

The theoretical volumetric fresh mixture flow rate into the engine, ${\dot{V}}_{\mathrm{t}}$, is

(19.13)$\begin{array}{cc}{\dot{V}}_{\text{t}}\left({\text{m}}^3\text{/s}\right)& =V_{\text{e}}\left({\text{m}}^3\text{/cycle}\right)\times cycles/revolution\\ & \times \mathrm{enginespeed}\text{(revolutions/s),}\end{array}$

where Ve is engine displacement, the number of cycles per revolution is 1/2 for a four-stroke engine, and the engine speed (revolutions per second) is ωe/2, where ωe is the engine speed in radians per second. Therefore,

(19.14)$\begin{array}{c}{\dot{V}}_{\text{t}}=V_{\text{e}}\times \frac{1}{2}\times \frac{\omega }{2\pi }=\frac{V_{\text{e}}{\omega }_{\text{e}}}{4\pi }.\end{array}$

This model assumes that the air is an ideal gas. According to the ideal gas law,

(19.15)$\begin{array}{c}pV=\frac{m}{m^{\text{'}}}RT,\end{array}$

where p is the absolute pressure, V is the volume of the vessel containing the gas, m is the mass of the gas, m′ is the molar mass of the gas, R is the gas constant, and T is the temperature in kelvins. Therefore, $m=\frac{p{m}^{\text{'}}V}{RT}$, and the density of the gas in the vessel is

(19.16)$\begin{array}{c}\rho =\frac{m}{V}=\frac{p{m}^{\text{'}}}{RT}=\frac{p}{R_{\text{a}}T},\end{array}$

where Ra = R/m′ and for air Ra ≈ 287 N m/kg/K. Further, the mass air flow rate, $\dot{m}$, as a function of the volumetric air flow rate, $\dot{V}$, is

(19.17)$\begin{array}{c}\dot{m}=\frac{p{m}^{\text{'}}}{RT}\dot{V}=\frac{p}{R_{\text{a}}T}\dot{V}.\end{array}$

For an engine, $\dot{V}$ is replaced by ${\dot{V}}_t$ and the speed of air flow is $v=\dot{V}/A$, where A is the cross-sectional area of any point in the intake manifold. The constriction in the manifold is the throttle, whose cross-sectional area is θ × A, where θ is percent of throttle opening. So the mass flow rate of air entering the engine is

(19.18)$\begin{array}{c}\dot{m}=\frac{p}{R_{\text{a}}T}{\dot{V}}_{\text{t}}=\frac{p}{R_{\text{a}}T}vA.\end{array}$

According to compressible fluid mechanics [104], the speed of air flow, v, is related to a Mach number, Ma, which is the ratio of air flow speed to sound speed $v_s=\sqrt{k{R}_{a}T}$—that is,

(19.19)$\begin{array}{c}{M}_{\text{a}}=\frac{v}{v_{\text{s}}}=\frac{v}{\sqrt{k{R}_{\text{a}}T}}=\frac{{\dot{V}}_{\text{t}}}{A\sqrt{k{R}_{\text{a}}T}},\end{array}$

where k is the specific heat ratio. Assume the stagnation state (where the flow is brought into a complete motionless condition in an isentropic process without other forces) holds. With the stagnation state for the ideal gas model in Sections 4.1 and 4.2 in Ref. [104], Equation 19.18 can be translated to

(19.20)$\begin{array}{c}\dot{m}=A\left(\frac{\sqrt{k}{M}_{\text{a}}{p}_0}{\sqrt{R_{\text{a}}}{T}_0}\right){\left(1+\frac{k-1}{2}{M}_{\text{a}}^2\right)}^{-\frac{k+1}{2(k-1)}},\end{array}$

where p0 and T0 are the stagnation pressure and temperature, respectively. Plugging 19.19 into 19.20 yields

(19.21)$\begin{array}{c}\dot{m}=A\left(\frac{{\dot{V}}_{\text{t}}{p}_0}{A{R}_{\text{a}}{T}_0}\right){\left(1+\frac{{\dot{V}}_{\text{t}}^2(k-1)}{2A^{2}k{R}_{\text{a}}{T}_0}\right)}^{-\frac{k+1}{2(k-1)}}.\end{array}$

Notice that Equations 19.20 and 19.21 apply to flow everywhere. When the flow is choked (i.e., Ma = 1) and the stagnation conditions (i.e., temperature, pressure) do not change, Equation 19.20 reduces to

(19.22)$\begin{array}{c}\dot{m}=A\left(\frac{\sqrt{k}{p}_0}{\sqrt{R_{\text{a}}}{T}_0}\right){\left(1+\frac{k-1}{2}\right)}^{-\frac{k+1}{2(k-1)}}.\end{array}$

For exact stoichiometric air-fuel ratio λ, fuel energy density Ef, and engine thermal efficiency η, the power developed by the engine is

(19.23)$\begin{array}{c}P=\lambda E_{\text{f}}\eta \left[A\left(\frac{{\dot{V}}_{\text{t}}{p}_0}{A{R}_{\text{a}}{T}_0}\right){\left(1+\frac{{\dot{V}}_{\text{t}}^2(k-1)}{2A^{2}k{R}_{\text{a}}{T}_0}\right)}^{-\frac{k+1}{2(k-1)}}\right].\end{array}$

Plugging in equation 19.14, we obtain

(19.24)$\begin{array}{c}P=\lambda E_{\text{f}}\eta \left[A\left(\frac{V_{\text{e}}{\omega }_{\text{e}}{p}_0}{4\pi A{R}_{\text{a}}{T}_0}\right){\left(1+\frac{V_{\text{e}}^2{\omega }_{\text{e}}^2(k-1)}{32{\pi }^2{A}^{2}k{R}_{\text{a}}{T}_0}\right)}^{-\frac{k+1}{2(k-1)}}\right].\end{array}$

$\begin{array}{l}\hfill P=\lambda E_{\text{f}}\eta \left[A\left(\frac{V_{\text{e}}{\omega }_{\text{e}}{p}_0}{4\mathrm{\pi }A{R}_{\text{a}}{T}_0}\right){\left(1+\frac{V_{\text{e}}^2{\omega }_{\text{e}}^2(k-1)}{32{\mathrm{\pi }}^2{A}^{2}k{R}_{\text{a}}{T}_0}\right)}^{-\frac{k+1}{2(k-1)}}\right].\end{array}$

The torque that the engine develops is

(19.25)$\begin{array}{c}\Gamma =\lambda E_{\text{f}}\eta \left[A\left(\frac{V_{\text{e}}{p}_0}{4\pi A{R}_{\text{a}}{T}_0}\right){\left(1+\frac{V_{\text{e}}^2{\omega }_{\text{e}}^2(k-1)}{32{\pi }^2{A}^{2}k{R}_{\text{a}}{T}_0}\right)}^{-\frac{k+1}{2(k-1)}}\right].\end{array}$

Empirical comparison shows that this model explains engine performance quite well up to peak torque and power. However, there are considerable differences between the model and the empirical engine curves after peak torque and power. Therefore, the engine model is modified by the addition of a correction term:

(19.26)$\begin{array}{c}P=\lambda E_{\text{f}}\eta \left[A\left(\frac{V_{\text{e}}{\omega }_{\text{e}}{p}_0}{4\pi A{R}_{\text{a}}{T}_0}\right){\left(1+\frac{V_{\text{e}}^2{\omega }_{\text{e}}^2(k-1)}{32{\pi }^2{A}^{2}k{R}_{\text{a}}{T}_0}\right)}^{-\frac{k+1}{2(k-1)}}\right]-\alpha P_{\text{max}}{\text{e}}^{\frac{\beta (\omega -{\omega }_{\text{p}})}{{\omega }_{\text{p}}}},\end{array}$

where α and β are coefficients to be calibrated. The specific form of the correction term is obtained mainly by trial and error from fitting a wide variety of engine power curves. This model, because of its simplicity, captures only the major aspect of an engine. Since many of the engine details are left out, the model exhibits only moderate accuracy even with the correction term.